Envelope Methods for Missing Responses in Multivariate Linear Models

Multivariate linear models are commonly used for modeling the relationships between multiple responses and covariates. With OLS estimators, one can interpret the results in the analysis. However, the coefficient OLS estimates sometimes suffer from large variability making it difficult to find statistical significance. By imposing the envelope structure in the response envelope model introduced by Cook et al. (2010), one can improve estimation efficiency by reducing substantial estimation variation of coefficient estimates. Although many scholars have developed envelope methods in recent years, an extension of envelope methods to allow for missing data is rarely explored in the literature. In this dissertation, we propose several approaches, the plug-in method (PI), the EM algorithm (EM), the Bayesian approach via data augmentation algorithm (DA), and the multiple imputation method (MI) for fitting envelope model to missing responses based on multivariate regression settings. In addition to the illustration of these approaches, we evaluate the performance among the approaches via a simulation study and conclude that the Bayesian data augmentation algorithm achieves the best performance across a variety of different Frequentist methods.